Regression-based weight generation algorithm in neural network for solution of initial and boundary value problems

This paper introduces a new algorithm for solving ordinary differential equations (ODEs) with initial or boundary conditions. In our proposed method, the trial solution of differential equation has been used in the regression-based neural network (RBNN) model for single input and single output syste...

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Veröffentlicht in:Neural computing & applications Jg. 25; H. 3-4; S. 585 - 594
Hauptverfasser: Chakraverty, S., Mall, Susmita
Format: Journal Article
Sprache:Englisch
Veröffentlicht: London Springer London 01.09.2014
Springer
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Abstract This paper introduces a new algorithm for solving ordinary differential equations (ODEs) with initial or boundary conditions. In our proposed method, the trial solution of differential equation has been used in the regression-based neural network (RBNN) model for single input and single output system. The artificial neural network (ANN) trial solution of ODE is written as sum of two terms, first one satisfies initial/boundary conditions and contains no adjustable parameters. The second part involves a RBNN model containing adjustable parameters. Network has been trained using the initial weights generated by the coefficients of regression fitting. We have used feed-forward neural network and error back propagation algorithm for minimizing error function. Proposed model has been tested for first, second and fourth-order ODEs. We also compare the results of proposed algorithm with the traditional ANN algorithm. The idea may very well be extended to other complicated differential equations.
AbstractList This paper introduces a new algorithm for solving ordinary differential equations (ODEs) with initial or boundary conditions. In our proposed method, the trial solution of differential equation has been used in the regression-based neural network (RBNN) model for single input and single output system. The artificial neural network (ANN) trial solution of ODE is written as sum of two terms, first one satisfies initial/boundary conditions and contains no adjustable parameters. The second part involves a RBNN model containing adjustable parameters. Network has been trained using the initial weights generated by the coefficients of regression fitting. We have used feed-forward neural network and error back propagation algorithm for minimizing error function. Proposed model has been tested for first, second and fourth-order ODEs. We also compare the results of proposed algorithm with the traditional ANN algorithm. The idea may very well be extended to other complicated differential equations.
Author Chakraverty, S.
Mall, Susmita
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  surname: Chakraverty
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  organization: Department of Mathematics, National Institute of Technology
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  givenname: Susmita
  surname: Mall
  fullname: Mall, Susmita
  organization: Department of Mathematics, National Institute of Technology
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Issue 3-4
Keywords Feed-forward model
Error back propagation
Regression fitting
Artificial neural network
Differential equation
Backpropagation
Input output
Initial condition
Initial value problem
Error function
Minimization
Regression analysis
Neural network
Boundary condition
Modeling
Weighted graph
Backpropagation algorithm
First order equation
Boundary value problem
Feedforward neural nets
Regression model
Feedforward
Growth of error
Mathematical programming
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PublicationTitle Neural computing & applications
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PublicationYear 2014
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Springer
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References SmaouiNAl-EneziSModelling the dynamics of nonlinear partial differential equations using neural networksJ Comput Appl Math200417027581049.65108207582310.1016/j.cam.2003.12.045
ShirvanyYHayatiMMoradianRMultilayer perceptron neural networks with novel unsupervised training method for numerical solution of the partial differential equationsAppl Soft Comput20099202910.1016/j.asoc.2008.02.003
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JckiewiezZRahamanMWelfentBDNumerical solution of a fredholm integra-differential equation modelling θ-neural networksAppl Math Comput200819525235363
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ZuradaJMIntroduction to artificial neural network1994EaganWest
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AarttLPVan der veerPNeural network method for solving partial differential equationsNeural Process Lett20011426127110.1023/A:1012784129883
MallSChakravertySRegression based neural network training for the solution of ordinary differential equationsInt J Math Model Numer Optim201341361491280.65081
DouglasJJonesBFPredictor–corrector methods for nonlinear parabolic differential equationsJ Ind Appl Math1963111952040116.0910415312310.1137/0111015
YazidHSPakdamanMModagheghHUnsupervised kernel least mean square algorithm for solving ordinary differential equationsNeurocomputing2011742062207110.1016/j.neucom.2010.12.026
McFallKSMahanJRArtificial neural network for solution of boundary value problems with exact satisfaction of arbitrary boundary conditionsIEEE Trans Neural Netw2009201221123310.1109/TNN.2009.2020735
Mai-DuyNTran-CongTNumerical solution of differential equations using multi quadric radial basis function networksNeural Netw20011418519910.1016/S0893-6080(00)00095-2
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HeSReifKUnbehauenRMultilayer neural networks for solving a class of partial differential equationsNeural Netw20001338539610.1016/S0893-6080(00)00013-7
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IG Tsoulos (1526_CR25) 2009; 72
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L Jianyu (1526_CR20) 2003; 16
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Z Jckiewiez (1526_CR21) 2008; 195
HJ Ricardo (1526_CR1) 2009
HS Yazid (1526_CR10) 2011; 74
JM Zurada (1526_CR29) 1994
IE Lagaris (1526_CR7) 1998; 9
L Manevitz (1526_CR17) 2005; 63
IG Tsoulos (1526_CR22) 2006; 7
KS McFall (1526_CR16) 2009; 20
A Malek (1526_CR9) 2006; 183
Y Shirvany (1526_CR12) 2009; 9
J Douglas (1526_CR3) 1963; 11
S Chakraverty (1526_CR28) 2009; 9
LP Aartt (1526_CR13) 2001; 14
S Haykin (1526_CR30) 1999
AJ Meade Jr (1526_CR6) 1994; 20
S Chakraverty (1526_CR27) 2006; 195
AJ Meade Jr (1526_CR5) 1994; 19
T Leephakpreeda (1526_CR18) 2002; 146
N Smaoui (1526_CR26) 2004; 170
IN Sneddon (1526_CR2) 2006
N Selvaraju (1526_CR11) 2010; 29
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References_xml – reference: LagarisIELikasACFotiadisDIArtificial neural networks for solving ordinary and partial differential equationsIEEE Trans Neural Netw19989987100010.1109/72.712178
– reference: SelvarajuNAbdul SamantJSolution of matrix Riccati differential equation for nonlinear singular system using neural networksInt J Comput Appl2010294854
– reference: HodaSANaglaHANeural network methods for mixed boundary value problemsInt J Nonlinear Sci2011113123162845978
– reference: LeephakpreedaTNovel determination of differential-equation solutions: universal approximation methodJ Comput Appl Math20021464434571013.65079192597210.1016/S0377-0427(02)00397-7
– reference: ZuradaJMIntroduction to artificial neural network1994EaganWest
– reference: DouglasJJonesBFPredictor–corrector methods for nonlinear parabolic differential equationsJ Ind Appl Math1963111952040116.0910415312310.1137/0111015
– reference: ManevitzLBitarAGivoliDNeural network time series forecasting of finite-element mesh adaptationNeurocomputing20056344746310.1016/j.neucom.2004.06.009
– reference: HaykinSNeural networks a comprehensive foundation1999Upper Saddle RiverPrentice Hall0934.68076
– reference: HeSReifKUnbehauenRMultilayer neural networks for solving a class of partial differential equationsNeural Netw20001338539610.1016/S0893-6080(00)00013-7
– reference: MeadeAJJrFernandezAAThe numerical solution of linear ordinary differential equations by feed forward neural networksMath Comput Model1994191250807.65079128417510.1016/0895-7177(94)90095-7
– reference: ChakravertySSinghVPSharmaRKRegression based weight generation algorithm in neural network for estimation of frequencies of vibrating platesJ Comput Methods Appl Mech Eng2006195419442021123.7405010.1016/j.cma.2005.08.008
– reference: ChakravertySSinghVPSharmaRKSharmaGKModelling vibration frequencies of annular plates by regression based neural networkAppl Soft Comput2009943944710.1016/j.asoc.2008.05.004
– reference: SneddonINElements of partial differential equations2006New YorkDover1115.35002
– reference: ParisiDRMarianiMCLabordeMASolving differential equations with unsupervised neural networksChem Eng Process20034271572110.1016/S0255-2701(02)00207-6
– reference: JianyuLSiweiLYingjianQYapingHNumerical solution of elliptic partial differential equation using radial basis function neural networksNeural Netw20031672973410.1016/S0893-6080(03)00083-2
– reference: MallSChakravertySRegression based neural network training for the solution of ordinary differential equationsInt J Math Model Numer Optim201341361491280.65081
– reference: ShirvanyYHayatiMMoradianRMultilayer perceptron neural networks with novel unsupervised training method for numerical solution of the partial differential equationsAppl Soft Comput20099202910.1016/j.asoc.2008.02.003
– reference: TsoulosIGLagarisIESolving differential equations with genetic programmingGenet Program Evolvable Mach20067335410.1007/s10710-006-7009-y
– reference: AarttLPVan der veerPNeural network method for solving partial differential equationsNeural Process Lett20011426127110.1023/A:1012784129883
– reference: McFallKSMahanJRArtificial neural network for solution of boundary value problems with exact satisfaction of arbitrary boundary conditionsIEEE Trans Neural Netw2009201221123310.1109/TNN.2009.2020735
– reference: MalekABeidokhtiSRNumerical solution for high order deferential equations, using a hybrid neural network—optimization methodAppl Math Comput20061832602711105.65340228280810.1016/j.amc.2006.05.068
– reference: TsoulosIGGavrilisDGlavasESolving differential equations with constructed neural networksNeurocomputing2009722385239110.1016/j.neucom.2008.12.004
– reference: SmaouiNAl-EneziSModelling the dynamics of nonlinear partial differential equations using neural networksJ Comput Appl Math200417027581049.65108207582310.1016/j.cam.2003.12.045
– reference: ReddyJNAn introduction to the finite element method1993New YorkMcGraw-Hill
– reference: MeadeAJJrFernandezAASolution of nonlinear ordinary differential equations by feed forward neural networksMath Comput Model19942019440818.65077130262910.1016/0895-7177(94)00160-X
– reference: YazidHSPakdamanMModagheghHUnsupervised kernel least mean square algorithm for solving ordinary differential equationsNeurocomputing2011742062207110.1016/j.neucom.2010.12.026
– reference: JckiewiezZRahamanMWelfentBDNumerical solution of a fredholm integra-differential equation modelling θ-neural networksAppl Math Comput200819525235363
– reference: LagarisIELikasACPapageorgiouDGNeural network methods for boundary value problems with irregular boundariesIEEE Trans Neural Netw2000111041104910.1109/72.870037
– reference: RicardoHJA modern introduction to differential equations20092AmsterdamElsevier1187.34001
– reference: Mai-DuyNTran-CongTNumerical solution of differential equations using multi quadric radial basis function networksNeural Netw20011418519910.1016/S0893-6080(00)00095-2
– volume: 170
  start-page: 27
  year: 2004
  ident: 1526_CR26
  publication-title: J Comput Appl Math
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Snippet This paper introduces a new algorithm for solving ordinary differential equations (ODEs) with initial or boundary conditions. In our proposed method, the trial...
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SubjectTerms Applied sciences
Artificial Intelligence
Computational Biology/Bioinformatics
Computational Science and Engineering
Computer Science
Computer science; control theory; systems
Connectionism. Neural networks
Data Mining and Knowledge Discovery
Exact sciences and technology
Image Processing and Computer Vision
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Original Article
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Probability and Statistics in Computer Science
Sciences and techniques of general use
Title Regression-based weight generation algorithm in neural network for solution of initial and boundary value problems
URI https://link.springer.com/article/10.1007/s00521-013-1526-4
Volume 25
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